Eigenvalues, geometry and instability in conservative models in applied mathematics.
应用数学保守模型中的特征值、几何和不稳定性。
基本信息
- 批准号:1211364
- 负责人:
- 金额:$ 21.8万
- 依托单位:
- 依托单位国家:美国
- 项目类别:Standard Grant
- 财政年份:2012
- 资助国家:美国
- 起止时间:2012-08-15 至 2016-07-31
- 项目状态:已结题
- 来源:
- 关键词:
项目摘要
Most of the equations derived to model physical phenomena support some formof special solutions such as standing or traveling waves, or other coherentstructures. These special solutions often correspond to real, observablephysical phenomena in the systems that these model equations are meant torepresent. One important aspect of this modeling is the stability of thesespecial solutions, as the stability of solutions determines if thesesolutions are likely to be physically observed. This project is aimed atunderstanding the stability and instability these coherent structures.This usually amounts to computing the index of the operator found bylinearizing about one of these coherent structures - that is the number ofeigenvalues of positive real part. This counts the dimension of theunstable manifold to the solution, giving valuable information on thedynamics of nearby solutions. While the emphasis of this project is onconservative systems we consider both conservative and dissipative systems.We use analytical, asymptotic and geometric techniques to identify unstableeigenvalues and to count the number of such eigenvalues.There are many mathematical models that are important for science andengineering that support solutions in the form of traveling waves. Oneexample is an equation known as the Korteweg-DeVries equation, an equationwhich governs the behavior of water in a narrow shallow body such as acanal. The Korteweg-DeVries equation has solutions called solitary waveswhich behave just as experience would suggest - they correspond to aquantity of water which propagates along without changing shape. Similarequations govern the propagation of light waves, propagation of adisturbance through a network such as the power grid, etc. It isimportant to understand the extent to which these solutions accuratelymodel the behavior of the underlying system. This is the question ofstability, which measures how robust these special solutions are. If aspecial solution like a traveling wave is stable it means that nearbysolutions behave in a similar way. This means such solutions are robust,and are likely to be observed: if the conditions are not exactly thosenecessary to produce a traveling wave but are close we expect to see asolution which is close to the special one. An unstable solution, on the other hand, is not robust. In order to observe these unstable solutionone must produce exactly the right conditions, which is very difficultto achieve in practice. This means that such solutions are more of mathematical interest than of physical importance. This project isprimarily concerned with developing mathematical techniques to understand the stability of traveling waves in a number of models including the Kuramoto model (a model for the behavior of power networks) and several nonlinear dispersive equations (which govern light waves in nonlinear media, water waves, waves in plasmas and many other phenomena).
大多数用于模拟物理现象的方程都支持某种形式的特殊解,如驻波或行波,或其他相干结构。这些特殊的解决方案往往对应于真实的,可观察到的物理现象的系统,这些模型方程是为了代表。这种建模的一个重要方面是这些特殊溶液的稳定性,因为溶液的稳定性决定了这些溶液是否可能被物理观察到。这个项目的目的是了解这些相干结构的稳定性和不稳定性,这通常相当于计算通过线性化这些相干结构中的一个而得到的算子的指数--即正真实的部分的特征值的数目。 这就计算了解的不稳定流形的维数,给出了关于附近解的动力学的有价值的信息。虽然这个项目的重点是保守系统,我们考虑保守和耗散系统。我们使用分析,渐近和几何技术来识别不稳定的特征值,并计算这样的特征值的数量。有许多数学模型是重要的科学和工程,支持行波形式的解决方案。一个例子是一个方程被称为Korteweg-DeVries方程,一个方程,它决定了水的行为在一个狭窄的浅体,如运河。Korteweg-DeVries方程的解称为孤立波,其行为与经验所建议的一样-它们对应于一定量的水,其传播沿着而不改变形状。类似的方程支配光波的传播,通过网络(如电网)传播干扰等。重要的是要了解这些解决方案在多大程度上准确地模拟了底层系统的行为。这就是稳定性的问题,它衡量这些特殊解决方案的鲁棒性。如果一个特殊的解决方案,如行波是稳定的,这意味着附近的解决方案的行为以类似的方式。这意味着这样的解决方案是强大的,并且很可能被观察到:如果条件不完全是产生行波所必需的,但很接近,我们希望看到一个接近特殊的解决方案。另一方面,不稳定的解决方案是不稳健的。为了观察这些不稳定的解决方案,一个人必须产生准确的正确的条件,这是非常难以实现的实践。这意味着这样的解决方案是更多的数学兴趣比物理的重要性。该项目主要关注开发数学技术,以了解许多模型中行波的稳定性,包括Kuramoto模型(电力网络行为的模型)和几个非线性色散方程(控制非线性介质中的光波,水波,等离子体中的波和许多其他现象)。
项目成果
期刊论文数量(0)
专著数量(0)
科研奖励数量(0)
会议论文数量(0)
专利数量(0)
数据更新时间:{{ journalArticles.updateTime }}
{{
item.title }}
{{ item.translation_title }}
- DOI:
{{ item.doi }} - 发表时间:
{{ item.publish_year }} - 期刊:
- 影响因子:{{ item.factor }}
- 作者:
{{ item.authors }} - 通讯作者:
{{ item.author }}
数据更新时间:{{ journalArticles.updateTime }}
{{ item.title }}
- 作者:
{{ item.author }}
数据更新时间:{{ monograph.updateTime }}
{{ item.title }}
- 作者:
{{ item.author }}
数据更新时间:{{ sciAawards.updateTime }}
{{ item.title }}
- 作者:
{{ item.author }}
数据更新时间:{{ conferencePapers.updateTime }}
{{ item.title }}
- 作者:
{{ item.author }}
数据更新时间:{{ patent.updateTime }}
Jared Bronski其他文献
Jared Bronski的其他文献
{{
item.title }}
{{ item.translation_title }}
- DOI:
{{ item.doi }} - 发表时间:
{{ item.publish_year }} - 期刊:
- 影响因子:{{ item.factor }}
- 作者:
{{ item.authors }} - 通讯作者:
{{ item.author }}
{{ truncateString('Jared Bronski', 18)}}的其他基金
Stability, Instability and Geometry in Applied Spectral Problems.
应用光谱问题中的稳定性、不稳定性和几何。
- 批准号:
1615418 - 财政年份:2016
- 资助金额:
$ 21.8万 - 项目类别:
Continuing Grant
Eigenvalue and Stability Problems in Applied Mathematics
应用数学中的特征值和稳定性问题
- 批准号:
0807584 - 财政年份:2008
- 资助金额:
$ 21.8万 - 项目类别:
Standard Grant
FRG: Collaborative Research in Semiclassical Asymptotic Questions in Integrable Nonlinear Wave Theory
FRG:可积非线性波理论中半经典渐近问题的合作研究
- 批准号:
0354462 - 财政年份:2004
- 资助金额:
$ 21.8万 - 项目类别:
Standard Grant
Mathematical Sciences: Postdoctoral Research Fellowship
数学科学:博士后研究奖学金
- 批准号:
9407473 - 财政年份:1994
- 资助金额:
$ 21.8万 - 项目类别:
Fellowship Award
相似国自然基金
2019年度国际理论物理中心-ICTP School on Geometry and Gravity (smr 3311)
- 批准号:11981240404
- 批准年份:2019
- 资助金额:1.5 万元
- 项目类别:国际(地区)合作与交流项目
新型IIIB、IVB 族元素手性CGC金属有机化合物(Constrained-Geometry Complexes)的合成及反应性研究
- 批准号:20602003
- 批准年份:2006
- 资助金额:26.0 万元
- 项目类别:青年科学基金项目
相似海外基金
Characterization of aneuploidy, cell fate and mosaicism in early development
早期发育中非整倍性、细胞命运和嵌合体的表征
- 批准号:
10877239 - 财政年份:2023
- 资助金额:
$ 21.8万 - 项目类别:
An ensemble deep learning model for tumor bud detection and risk stratification in colorectal carcinoma.
用于结直肠癌肿瘤芽检测和风险分层的集成深度学习模型。
- 批准号:
10564824 - 财政年份:2023
- 资助金额:
$ 21.8万 - 项目类别:
Characterization of aneuploidy, cell fate and mosaicism in early development
早期发育中非整倍性、细胞命运和嵌合体的表征
- 批准号:
10525693 - 财政年份:2022
- 资助金额:
$ 21.8万 - 项目类别:
Optimising the geometry of 3D printed steel structures against local and global instability.
优化 3D 打印钢结构的几何形状,以应对局部和全局的不稳定性。
- 批准号:
2593508 - 财政年份:2021
- 资助金额:
$ 21.8万 - 项目类别:
Studentship
Metastasis and biophysics of clusters of circulating tumor cells in the microcirculation
微循环中循环肿瘤细胞簇的转移和生物物理学
- 批准号:
9924267 - 财政年份:2018
- 资助金额:
$ 21.8万 - 项目类别:
Metastasis and biophysics of clusters of circulating tumor cells in the microcirculation
微循环中循环肿瘤细胞簇的转移和生物物理学
- 批准号:
10429911 - 财政年份:2018
- 资助金额:
$ 21.8万 - 项目类别: